Subgroups — C₂²×C₈×D₂₁₂ #28247925

Order 13568.

3694 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×D₅₃	C₂	1	2	yes	maximal, complemented
2	C₂²×C₄×D₂₁₂	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×D₂₁₂	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#13133129	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₂₁₂	C₄	1	1	yes	characteristic
8	C₂³×C₄×D₅₃	C₂²	1	2	yes	—
9	C₂³×#88322	C₂²	1	2	yes	—
10	C₂²×C₈×D₅₃	—	2	16	no	—
11	C₂²×C₈×D₅₃	C₂²	1	24	yes	complemented
12	C₂×C₄×D₂₁₂	C₄	1	12	yes	—
13	C₂×C₄×D₂₁₂	C₂²	1	12	yes	—
14	C₂²×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
15	C₂²×#11719921	C₄	1	2	yes	—
16	C₈×D₂₁₂	C₂²	1	64	yes	direct product factor
17	C₂×C₄⋊#88322	C₂²	1	12	yes	complemented
18	C₂×#13133129	C₂²	1	24	yes	complemented
19	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₃	C₂²	1	12	yes	complemented
23	C₂⁴×D₅₃	C₂×C₄	1	2	yes	—
24	C₂²×D₂₁₂	C₈	1	16	yes	direct product factor
25	C₂²×D₂₁₂	C₂×C₄	1	12	yes	—
26	C₂³×Dic₅₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×D₅₃	—	2	8	no	—
28	C₂²×C₄×D₅₃	—	2	8	no	—
29	C₂²×C₄×D₅₃	C₂×C₄	1	12	yes	—
30	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
31	C₂²×#88322	—	2	8	no	—
32	C₂²×#88322	C₂³	1	12	yes	—
33	C₂×C₈×D₅₃	—	2	96	no	—
34	C₂×C₈×D₅₃	C₂³	1	32	yes	complemented
35	C₄×D₂₁₂	C₂×C₄	1	48	yes	—
36	C₄×D₂₁₂	C₂³	1	16	yes	—
37	C₂×C₄⋊Dic₅₃	C₂×C₄	1	12	yes	—
38	C₂×#11719921	C₂×C₄	1	24	yes	—
39	C₄⋊#88322	C₂³	1	16	yes	complemented
40	#13133129	C₂³	1	32	yes	complemented
41	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	—	2	4	no	—
45	C₂²×C₈×C₅₃	D₄	1	8	yes	complemented
46	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
47	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
48	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
49	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
50	C₄×C₈×C₅₃	C₂³	1	16	yes	complemented
51	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂³×D₅₃	—	2	8	no	—
53	C₂³×D₅₃	—	2	8	no	—
54	C₂³×D₅₃	C₂×C₈	1	16	yes	complemented
55	C₂³×D₅₃	C₂²×C₄	1	12	yes	—
56	C₂²×C₄×C₅₃	—	2	4	no	—
57	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	complemented
58	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
62	C₂²×C₄×C₅₃	#41	1	4	yes	—
63	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
64	C₂²×Dic₅₃	—	2	8	no	—
65	C₂²×Dic₅₃	C₂²×C₄	1	12	yes	—
66	C₂×D₂₁₂	C₂×C₈	1	96	yes	direct product factor
67	C₂×D₂₁₂	C₂²×C₄	1	16	yes	—
68	C₂×C₄×D₅₃	—	2	48	no	—
69	C₂×C₄×D₅₃	—	2	24	no	—
70	C₂×C₄×D₅₃	—	2	24	no	—
71	C₂×C₄×D₅₃	C₂²×C₄	1	24	yes	—
72	C₂×C₄×D₅₃	C₂⁴	1	8	yes	—
73	C₂×C₈×C₅₃	—	2	12	no	—
74	C₂×C₈×C₅₃	C₂×D₄	1	24	yes	complemented
75	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
76	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
77	C₄²×C₅₃	C₂²×C₄	1	12	yes	—
78	C₄²×C₅₃	C₂⁴	1	4	yes	—
79	#11719921	C₂²×C₄	1	32	yes	—
80	C₄⋊Dic₅₃	C₂²×C₄	1	16	yes	—
81	C₂×#88322	—	2	24	no	—
82	C₂×#88322	C₂⁴	1	8	yes	—
83	C₈×D₅₃	—	2	64	no	—
84	C₂³×C₅₃	C₄×D₄	1	4	yes	—
85	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
86	C₂³×C₅₃	#131	1	4	yes	—
87	C₂³×C₅₃	C₂³×C₄	1	3	yes	—
88	C₂²×D₅₃	—	2	24	no	—
89	C₂²×D₅₃	—	2	24	no	—
90	C₂²×D₅₃	—	2	64	no	—
91	C₂²×D₅₃	C₂²×C₈	1	48	yes	complemented
92	C₂²×D₅₃	C₂³×C₄	1	8	yes	—
93	C₂×C₄×C₅₃	—	2	12	no	—
94	C₂×C₄×C₅₃	C₄×D₄	1	12	yes	—
95	C₂×C₄×C₅₃	C₂²×C₈	1	24	yes	complemented
96	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
97	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
98	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
99	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
100	C₂×C₄×C₅₃	C₂×#41	1	6	yes	—
101	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
102	C₂×Dic₅₃	—	2	24	no	—
103	C₂×Dic₅₃	C₂³×C₄	1	8	yes	—
104	D₂₁₂	C₂²×C₈	1	64	yes	direct product factor
105	C₄×D₅₃	—	2	48	no	—
106	C₄×D₅₃	—	2	8	no	—
107	C₄×D₅₃	—	2	8	no	—
108	C₈×C₅₃	—	2	4	no	—
109	C₈×C₅₃	C₂²×D₄	1	8	yes	complemented
110	#88322	—	2	8	no	—
111	C₂²×C₈×D₄	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	C₈×D₄	1	16	yes	complemented
113	C₂²×C₅₃	C₂×C₄×D₄	1	6	yes	—
114	C₂²×C₅₃	C₂³×C₈	1	6	yes	—
115	C₂²×C₅₃	C₂×#131	1	6	yes	—
116	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
117	C₂×D₅₃	—	2	8	no	—
118	C₂×D₅₃	—	2	96	no	—
119	C₂×D₅₃	—	2	8	no	—
120	C₂×D₅₃	C₂³×C₈	1	16	yes	complemented
121	C₄×C₅₃	—	2	4	no	—
122	C₄×C₅₃	C₂×C₄×D₄	1	6	yes	—
123	C₄×C₅₃	C₂³×C₈	1	8	yes	complemented
124	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
125	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
126	Dic₅₃	—	2	8	no	—
127	C₂⁴×C₈	—	53	2	no	—
128	C₂²×C₄×D₄	—	53	1	no	—
129	C₂×C₈×D₄	—	53	24	no	—
130	C₂²×C₄⋊C₈	—	53	1	no	—
131	C₂²×C₂²⋊C₈	—	53	2	no	—
132	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic, complemented
133	C₂×C₅₃	C₂×C₈×D₄	1	12	yes	complemented
134	C₂×C₅₃	C₂²×C₄×D₄	1	1	yes	characteristic
135	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
136	C₂×C₅₃	C₂²×#131	1	1	yes	characteristic
137	D₅₃	—	2	16	no	—
138	C₂³×D₄	—	53	1	no	—
139	C₂⁴×C₄	—	53	2	no	—
140	C₂³×C₈	—	53	2	no	—
141	C₂³×C₈	—	53	24	no	—
142	C₂³×C₈	—	106	16	no	—
143	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
144	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
145	C₂×C₄×D₄	—	53	12	no	—
146	C₂×C₄×D₄	—	53	12	no	—
147	C₂²×C₄⋊C₄	—	53	1	no	—
148	C₂²×C₂²⋊C₄	—	53	2	no	—
149	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
150	C₈×D₄	—	53	64	no	—
151	C₂×C₄⋊C₈	—	53	12	no	—
152	C₂×C₂²⋊C₈	—	53	24	no	—
153	C₂×C₄×C₈	C₂×D₅₃	1	12	yes	complemented
154	C₅₃	C₂²×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
155	C₂⁵	—	53	2	no	—
156	C₂²×D₄	—	53	12	no	—
157	C₂²×D₄	—	53	16	no	—
158	C₂³×C₄	—	53	2	no	—
159	C₂³×C₄	—	53	12	no	—
160	C₂³×C₄	—	53	12	no	—
161	C₂³×C₄	—	106	8	no	—
162	C₂³×C₄	—	106	8	no	—
163	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
164	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
165	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
166	C₂²×C₈	—	2	4	no	—
167	C₂²×C₈	—	53	12	no	—
168	C₂²×C₈	—	53	32	no	—
169	C₂²×C₈	—	106	8	no	—
170	C₂²×C₈	—	106	96	no	—
171	C₂²×C₈	D₂₁₂	1	8	yes	direct product factor
172	C₂²×C₈	C₂²×D₅₃	1	6	yes	—
173	C₂²×C₈	C₂²×D₅₃	1	6	yes	—
174	C₄×D₄	—	53	48	no	—
175	C₄×D₄	—	53	16	no	—
176	C₂×C₄⋊C₄	—	53	12	no	—
177	C₂×C₂²⋊C₄	—	53	24	no	—
178	C₂×C₄²	C₄×D₅₃	1	6	yes	—
179	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
180	C₄⋊C₈	—	53	16	no	—
181	C₂²⋊C₈	—	53	32	no	—
182	C₄×C₈	C₂²×D₅₃	1	16	yes	complemented
183	C₂⁴	—	53	16	no	—
184	C₂⁴	—	53	12	no	—
185	C₂⁴	—	106	8	no	—
186	C₂⁴	—	106	8	no	—
187	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
188	C₂×D₄	—	53	96	no	—
189	C₂×D₄	—	53	16	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	53	12	no	—
192	C₂²×C₄	—	53	24	no	—
193	C₂²×C₄	—	53	8	no	—
194	C₂²×C₄	—	106	8	no	—
195	C₂²×C₄	—	106	48	no	—
196	C₂²×C₄	—	106	24	no	—
197	C₂²×C₄	—	106	24	no	—
198	C₂²×C₄	C₈×D₅₃	1	8	yes	complemented
199	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
200	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
201	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
202	C₂²×C₄	C₂×D₂₁₂	1	4	yes	—
203	C₂²×C₄	#11719945	1	4	yes	—
204	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
205	C₂×C₈	—	2	12	no	—
206	C₂×C₈	—	53	8	no	—
207	C₂×C₈	—	106	24	no	—
208	C₂×C₈	—	106	64	no	—
209	C₂×C₈	C₂×D₂₁₂	1	24	yes	direct product factor
210	C₂×C₈	C₂³×D₅₃	1	4	yes	—
211	C₂×C₈	C₂³×D₅₃	1	4	yes	—
212	C₄⋊C₄	—	53	16	no	—
213	C₂²⋊C₄	—	53	32	no	—
214	C₄²	C₂×C₄×D₅₃	1	12	yes	—
215	C₄²	C₂³×D₅₃	1	4	yes	—
216	C₂³	—	53	48	no	—
217	C₂³	—	53	8	no	—
218	C₂³	—	106	24	no	—
219	C₂³	—	106	24	no	—
220	C₂³	—	106	64	no	—
221	C₂³	C₄×D₂₁₂	1	4	yes	—
222	C₂³	C₂×C₈×D₅₃	1	4	yes	—
223	C₂³	#13133223	1	4	yes	—
224	C₂³	C₂²×C₄×D₅₃	1	3	yes	—
225	D₄	—	53	64	no	—
226	C₂×C₄	—	2	12	no	—
227	C₂×C₄	—	53	8	no	—
228	C₂×C₄	—	106	24	no	—
229	C₂×C₄	—	106	48	no	—
230	C₂×C₄	—	106	8	no	—
231	C₂×C₄	—	106	8	no	—
232	C₂×C₄	C₄×D₂₁₂	1	12	yes	—
233	C₂×C₄	C₂×C₈×D₅₃	1	24	yes	complemented
234	C₂×C₄	C₂²×C₄×D₅₃	1	4	yes	—
235	C₂×C₄	C₂²×C₄×D₅₃	1	3	yes	—
236	C₂×C₄	C₂²×C₄×D₅₃	1	4	yes	—
237	C₂×C₄	C₂²×D₂₁₂	1	6	yes	—
238	C₂×C₄	C₂×#11719945	1	6	yes	—
239	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
240	C₈	—	2	4	no	—
241	C₈	—	106	8	no	—
242	C₈	C₂²×D₂₁₂	1	8	yes	direct product factor
243	C₂²	—	53	16	no	—
244	C₂²	—	106	8	no	—
245	C₂²	—	106	8	no	—
246	C₂²	—	106	96	no	—
247	C₂²	C₈×D₂₁₂	1	16	yes	direct product factor
248	C₂²	C₂×C₄×D₂₁₂	1	6	yes	—
249	C₂²	C₂²×C₈×D₅₃	1	6	yes	—
250	C₂²	C₂×#13133223	1	6	yes	—
251	C₂²	C₂³×C₄×D₅₃	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	106	8	no	—
254	C₄	C₂×C₄×D₂₁₂	1	6	yes	—
255	C₄	C₂²×C₈×D₅₃	1	8	yes	complemented
256	C₄	C₂³×D₂₁₂	1	1	yes	characteristic
257	C₄	C₂²×#11719945	1	1	yes	characteristic
258	C₂	—	106	16	no	—
259	C₂	C₂×C₈×D₂₁₂	1	12	yes	direct product factor
260	C₂	C₂²×C₄×D₂₁₂	1	1	yes	characteristic
261	C₂	C₂³×C₈×D₅₃	1	1	yes	characteristic
262	C₂	C₂²×#13133223	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂²×C₄
prime radical	C₂³×D₂₁₂	C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂²×C₈×D₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂³×C₈×D₅₃	C₂	semidirect
3. C₂×C₈×D₂₁₂	C₂	direct (N × H)
4. C₂²×C₄⋊#88322	C₂	semidirect
5. C₂²×#13133129	C₂	semidirect
6. C₂²×C₄×C₈×C₅₃	C₂	semidirect
11. C₂²×C₈×D₅₃	C₂²	semidirect
16. C₈×D₂₁₂	C₂²	direct (N × H)
17. C₂×C₄⋊#88322	C₂²	semidirect
18. C₂×#13133129	C₂²	semidirect
22. C₂×C₄×C₈×C₅₃	C₂²	semidirect
24. C₂²×D₂₁₂	C₈	direct (N × H)
34. C₂×C₈×D₅₃	C₂³	semidirect
39. C₄⋊#88322	C₂³	semidirect
40. #13133129	C₂³	semidirect
45. C₂²×C₈×C₅₃	D₄	semidirect
50. C₄×C₈×C₅₃	C₂³	semidirect
54. C₂³×D₅₃	C₂×C₈	semidirect
57. C₂²×C₄×C₅₃	C₂×C₈	semidirect
66. C₂×D₂₁₂	C₂×C₈	direct (N × H)
74. C₂×C₈×C₅₃	C₂×D₄	semidirect
91. C₂²×D₅₃	C₂²×C₈	semidirect
95. C₂×C₄×C₅₃	C₂²×C₈	semidirect
104. D₂₁₂	C₂²×C₈	direct (N × H)
109. C₈×C₅₃	C₂²×D₄	semidirect
112. C₂²×C₅₃	C₈×D₄	semidirect
120. C₂×D₅₃	C₂³×C₈	semidirect
123. C₄×C₅₃	C₂³×C₈	semidirect
132. C₂²×C₄×C₈	D₅₃	semidirect
133. C₂×C₅₃	C₂×C₈×D₄	semidirect
153. C₂×C₄×C₈	C₂×D₅₃	semidirect
154. C₅₃	C₂²×C₈×D₄	semidirect
171. C₂²×C₈	D₂₁₂	direct (N × H)
182. C₄×C₈	C₂²×D₅₃	semidirect
198. C₂²×C₄	C₈×D₅₃	semidirect
209. C₂×C₈	C₂×D₂₁₂	direct (N × H)
233. C₂×C₄	C₂×C₈×D₅₃	semidirect
242. C₈	C₂²×D₂₁₂	direct (N × H)
247. C₂²	C₈×D₂₁₂	direct (N × H)
255. C₄	C₂²×C₈×D₅₃	semidirect
259. C₂	C₂×C₈×D₂₁₂	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₈×D₅₃	2
C₂²×C₄×D₂₁₂	2
C₂×C₈×D₂₁₂	2
C₂²×C₄⋊#88322	2
C₂²×#13133129	2
C₂²×C₄×C₈×C₅₃	2
C₂²×C₈×D₄	53

Subgroups — C22×C8×D212 #28247925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×D₂₁₂ #28247925